Resting Stages and the Population Dynamics of Harmful Algae in ...

Table 1: NotationVariables Meaning UnitsM Motile algae cells/mLN Non-motile algae cells/mLR Nutrient concentration µmol/LParameters MeaningUnitsK Half-saturation constant for algal growth µmol/Lµ max maximal growth rate of motile algae day −1δ Rate of conversion from motile to non-motile day −1γ Rate of conversion from non-motile to motile day −1q Nutrient quota of algae µmol/cellR in Nutrient supply µmol/LdMdt= µ maxRMK +R−δM +γN,dNdtdRdt= δM −γN,= − µ maxRMK +R q. (1)Each differential equation represents the factors we assume to impact thechange in populations and toxicity. We assume that only motile algae (M) cantake up the nutrient (R) and reproduce; therefore there exists a relationshipbetween the growth of the motile population and the nutrient. It has been wellestablished that the growthof the algae is limited by the nutrient concentration,and the Monod function,µ max RMK +R ,has repeatedly been used to represent the interaction between the taking up ofnutrient and the growth of P. parvum [3]. In our model µ max (day −1 ) is themaximal growth rate of the motile algae and K (µmol/L) is the half saturationconstant. The consumption of the nutrient represented in dR is proportional todtthe growth rate of the motile algae in dM with coefficient q (µmol/cell), whichdtconverts the units from cells to nutrients (µmol). We also assume that motilealgae converts to non-motile algae (N) at a constant rate, δ (day −1 ), and thatnon-motile algae converts to motile algae at a constant rate, γ (day −1 ).2.1 Reduction of OrderThe system is changed from three differential equations to a much simpler systemof only two differential equations. T is the total nutrients in the system3